Compactness and fractal dimensions of inhomogeneous continuum random trees

TL;DR

This paper introduces a novel stick-breaking construction for inhomogeneous continuum random trees, proving a conjecture on their compactness and calculating their fractal dimensions, advancing understanding of their geometric properties.

Contribution

It presents a new construction method for ICRTs and confirms a conjecture on their compactness, also providing fractal dimension calculations.

Findings

Established necessary and sufficient conditions for compactness of ICRTs.

Derived explicit formulas for Minkowski, Packing, and Hausdorff dimensions.

Validated the new construction by comparison with Lévy trees.

Abstract

We introduce a new stick-breaking construction for inhomogeneous continuum random trees (ICRT). This new construction allows us to prove the necessary and sufficient condition for compactness conjectured by Aldous, Miermont and Pitman arXiv:math/0401115 by comparison with L\'evy trees. We also compute the fractal dimensions (Minkowski, Packing, Hausdorff).